Projectivity and flatness over the endomorphism ring of a finitely generated $(H,\mathcal{C})$-comodule

Let $k$ be a commutative ring, $H$ a bialgebra over $k$, $A$ an $H$-algebra and $\mathcal{C}$ an $(H,A)$-coring, i.e a left $(H,A)$-bimodule which is an $A$-coring in a compatible way. Left $(H,A)$-bimodules and their deformations play a fundamental role in the study of the differential geometry of noncommutative manifolds. Let $\mathrm{\Lambda }$ be a right $(H,\mathcal{C})$-comodule. The $k$-module ${}_{H}En{d}^{\mathcal{C}}(\mathrm{\Lambda })$ of right $(H,\mathcal{C})$-colinear maps from $\mathrm{\Lambda }$ to $\mathrm{\Lambda }$ is a ring. Let us assume that $\mathcal{C}$ is flat as a left $A$-module. If $\mathrm{\Lambda }$ is finitely generated (finitely presented) as a right $(H,\mathcal{C})$-comodule, we give necessary and sufficient conditions for projectivity and flatness of a module over ${}_{H}En{d}^{\mathcal{C}}(\mathrm{\Lambda })$. If $\mathcal{C}$ contains a fixed $H$-grouplike element, we can replace $\mathrm{\Lambda }$ with $A$.